We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The Bar-Natan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable Levine--Tristram signature.